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Tuesday, October 25, 2011

Have you ever collected leaves? There come in so many different shapes. Some are long and slender, some roundish and then some are heart shaped. Some plants even have leaves that have deep splits in them, as if the leaf has been torn.

Isn't it amazing that they come in so many different shapes? What would have affected their shape? First of all, why is the average leaf shaped to be thick in the middle and tapered to the ends?

Leaf is the kitchen of the plant. That's where the plant cooks its food. To cook its food, it needs water, minerals, air and sunlight. Water and minerals are transported from the roots through a network of viens. It also absorbs carbon dioxide from the surrounding air and sunlight during the day. The shape of a leaf has been designed by nature to be as efficient as possible in doing its job! The difference in the shapes are a result of the different kind of environments each of the plants are designed to live in. Let us examine a few such factors.

Plants face a strange predicament. While water and minerals are absorbed closer to the ground, light is available higher up from the ground. Forest floors are usually shaded by trees that are higher. A plant that is not tall and has to grow in shade, needs to capture as much sunlight as possible. Such plants develop fat roundish leaves. But it's not easy for the tall plants either. They have to transport water and minerals all the way from the ground to the leaves. The farther the veins that carry water go, the narrower they become. Such narrow viens can not form intricate network of viens. Tall plants therefore have narrower leaves with mostly straight viens.

As the plant grows, the leaves at the bottom of the plant become less effective as they are shaded by other leaves. The plant therefore lets go or sheds the lower leaves. So leaves need to be easily detachable. Leaves also need to be flexible and must bend easily to give way to passing wind. Otherwise wind force will probably uproot the tree. Therefore leaves are slender at the point they are attached to the stem, to make them flexible and easily detachable.

Big leaves also have the problem of tearing up in strong winds. Plants that have to survive windy places, like the coconut tree, develop cuts in their leaves so that air can pass through easily. Some such plants also have thread like leaves, like the pine tree.

Water is scarce in some places. Some plants that grow in drier places must grab and store as much water as possible when they can. Such plants have thicker leaves that store water. They also tend to have smaller leaves to avoid the stored water from evaporating too fast.Even an abundance of water needs to be tackled. Plants that grow near or on water need to keep their leaves above water where they can have enough air and light. Such plants also have thicker leaves that have air pockets to keep them afloat.

Plants that grow in dry and cold places usually have thin leaves. Such leaves minimize water evaporation from the leaves. They also do not let ice/frost form easily on them, protecting the plant from damage.

Apart from these simple factors, there are many other reasons why plants may develop different style of leaves. It is a combination of circumstances that finally decide the best leaf shape for a plant. While most simple leaf shapes can be explained, certain shapes have been a matter of debate for scientists since a long time.

Friday, October 7, 2011

It was a dark night. Hedwig, the owl, was sitting up on the branch of a tree when he spotted a mouse on the ground. He swoops down silently to catch the mouse. Owls can be really silent when they just glide down. What path should he take to reach the mouse? Should it be a straight line? Or a curve? What curve?

It turns out, that Hedwig the owl should follow a curve called "Cycloid" to reach the mouse the fastest. Birds also tend to take this path when they glide short distances from branch to branch.

There is one more interesting property of a cycloid. If a smooth slide is made into the shape of a cycloid, no matter at what point you jump on to the slide, you'll take the same amount of time to reach the bottom. Isn't that interesting?

Cycloid is the path a point on circle takes when it rolls on a surface. It looks simple, but has many variations. When we roll the circle over another circle, instead of a straight line, we get a much cooler looking Epicycloid. And if the circle moves around the inner circumference, rather than the outer circumference, we get a Hypocycloid.

Requires Java permissions to run.

Play around with the sliders above. Can you get the cycloid to trace a straight line? It happens when the small circle is just half the radius of the large circle and is moving inside the large circle.

If you observe a point inside the circle, rather than one on the circumference, it appears to move slightly differently. That's called a Trochoid. So we can have Epitrochoids and Hypotrochoids from such points. Actually cycloids are a special type of trochoid where the point is right on the edge of the circle.

Tuesday, October 4, 2011

The rose curve looks like a petalled flower! Sometimes it is also called the rhodonea curve. It was discovered by Guido Grandi, an Italian mathematician. It is a really simple curve. It is just a regular sine curve (what a wave looks like) plotted on a polar graph.

Now what's a polar graph? A regular graph has two axes - one horizontal (X) axis and a vertical (Y) axis. A point on this (XY) graph is represented as the combination of two values (x, y) value. To reach such a point you walk x distance on the horizontal (X) axis and then walk y distance parallel to Y axis. A polar graph, on the other hand, describes points as the combination of an angle (θ) and distance. (r). How to reach a point on a polar graph? You start facing east, turn anti clockwise by angle θ and then walk r distance.

On a polar curve, if we change the distance (r) for every angle, we get some interesting plots. Rose curve is one such plot where the distance for any angle is related to the angle through a wave function. That is, as the angle changes, the distance increases gradually to a maximum value and then decreases to a minimum. Depending on how fast the wave function oscillates (pitch or frequency), we get different numbers of rise and fall.

Below is an interactive applet illustrating the rose curve (r = cos(slider1 * θ / slider2)). It does create some beautiful flower patterns. Pause the animation (button at bottom left corner) and change the sliders yourself to observe how the flower changes pattern. Particularly observe the following two curious behaviors:

Set the second slider to 1. Change the first slider and count the number of petals. Do you see a pattern? Number of petals is same as the number when the number is odd. But it is double of the number when the number is even.

Set both the sliders to the same value. It's a circle!

We can do many more interesting patterns by playing around with this. In another example below, the value is determined by applying the wave function twice (r = cos(k * sin(θ))

Another interesting pattern made using the Rose curve is the Maurer Rose. In addition to the plain rose curve, it has straight lines joining two points on the curve at a certain distance. It gives an interesting 3D pattern to the curve. Play with the interactive applet below to create some interesting Maurer Roses. Set the thickness to a larger value to see the Rose curve on which the Maurer Rose is based on.

Rose curves are used in practice to describe shapes of various electrical and magnetic fields.

Few other points to ponder on:

Imagine why the number of petals changes with change of wave frequency.

Experiment with the sliders of the Maurer Rose applet above, particularly the one marked D. Looking at the pattern, can you tell when D is a prime number?